The Cost of Free Will: Directed Information and the Physics of Choice

Humanity’s Oldest Debate

Do you have free will?

For thousands of years, this question has torn apart philosophers, theologians, and scientists:

Determinists say: The universe is a causal chain, each event determined by previous events. Your “choices” are just results of brain neurons operating according to physical laws—freedom is an illusion.

Free will advocates say: Humans have real ability to choose, otherwise morality, responsibility, meaning all lose foundation. Freedom is human essence.

Compatibilists say: Freedom and determinism can coexist—as long as you act “according to your will,” that’s freedom, regardless of whether will itself is determined.

In 2024, physics gives a fourth answer:

Free will is not a metaphysical concept, but a measurable physical quantity—it’s the directed information your actions produce on the future, and this has physical lower bounds and energy costs.

Freedom is not all-or-nothing, but degree—and this degree can be calculated with information theory.

Act I: Definition of Freedom

What is freedom?

Problems with Naive Definitions

“Can do what I want” — But where does “want” come from? If “want” is determined by brain state, is this freedom?

“Not forced by external forces” — But genes, education, culture are all “external forces,” can you escape them?

“Can choose otherwise” — But under same initial conditions, physical laws give same results, where’s “otherwise”?

Information-Theoretic Definition

GLS framework gives operational definition:

Degree of freedom = How much distinguishable influence your actions can have on the future

Formalized:

$$\text{Freedom}=I(M^n\to Y^n)$$

where:

• $M^n=(M_1,M_2,\dots ,M_n)$: Your action sequence
• $Y^n=(Y_1,Y_2,\dots ,Y_n)$: Future you observe
• $I(M^n\to Y^n)$: Directed information

What Is Directed Information?

Definition:

$$I(M^n\to Y^n)=\sum _{i=1}^{n}I(M^i;Y_i|Y^{i-1})$$

It measures: Given past observations, how much new information do your actions provide about current observation.

In plain language:

• If your actions completely don’t affect future, $I=0$ (no freedom)
• If your actions significantly change future, $I$ is large (high freedom)

Freedom = information measure of causal influence.

Act II: Physical Lower Bounds of Freedom

Now key question: Does freedom have lower bounds?

Answer: Yes, and profound.

Condition 1: Directed Information Lower Bound

To have free will, must have:

$$I(M^n\to Y^n)>c_1>0$$

Your actions must be able to affect future, and this influence is measurable.

If $I=0$, means:

• Your actions are random noise (unrelated to future)
• Or your actions are completely passive (determined by outside)

No influence = no freedom.

Condition 2: Empowerment Lower Bound

Define Empowerment:

$${\mathcal{E}}_k=\underset{p(a^k)}{\max }I(A^k;S_{k+1})$$

It measures: Your future $k$ steps of actions, maximum influence on state.

To have freedom, need:

$${\mathcal{E}}_k>c_2>0$$

You must have potential choice space—can do different things.

Condition 3: Energy-Information Trade-off

Most profound constraint comes from thermodynamics:

$$⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$$

This is corollary of Jarzynski-Sagawa-Ueda equality:

Information $I$ you acquire can reduce need for work $W$, but has cost.

From another angle:

$$⟨W⟩+k_{B}T⟨I⟩\ge \Delta F$$

Freedom requires energy: either consume physical work $W$, or consume information processing cost $k_{B}TI$.

Act III: The Cost of Freedom

Freedom is not free.

Brain’s Energy Consumption

Human brain:

• Mass: ~1.4kg (2% of body weight)
• Energy consumption: ~20W (20% of total metabolism)

Why so energy-intensive?

Because brain is:

1. Processing information (neuron firing, synaptic transmission)
2. Maintaining memory (synaptic weight stabilization)
3. Making decisions (evaluating options, choosing actions)

These are all information operations, and information has energy cost.

Landauer Limit

Minimum energy to erase 1 bit of information:

$$E_{\min }=k_{B}T\ln 2\approx 3\times 10^{-21}\text{ J}$$

(at room temperature)

Brain processes about $10^{16}$ synaptic events per second, if each involves 1 bit, total energy:

$$E\sim 10^{16}\times 3\times 10^{-21}\approx 30\text{ mW}$$

Actual consumption 20W, meaning efficiency about 0.15%—most energy used maintaining structure, fighting entropy increase.

Physical cost of free will: 20% of brain’s energy budget.

Cost of Decision-Making

Making decisions consumes more energy than execution.

Experiments:

• Simple reaction (press button when see light): Few brain regions activated
• Complex choice (weighing multiple options): Many brain regions activated, high energy consumption

Complex decisions consume energy several times that of simple reactions.

This is why:

• Decision fatigue is real (consecutive decisions deplete glucose)
• Daydreaming is easier (reduces information processing intensity)
• Automated habits save effort (reduces active decisions)

Freedom is expensive—you must pay cost for each choice.

Act IV: Limitations of Determinism

Free will advocates often encounter a rebuttal:

“Laplace’s Demon” — If there’s an omniscient demon knowing all particles’ positions and velocities in universe, it can predict everything (including your “choices”). So freedom is illusion.

Quantum Uncertainty

First limitation: Heisenberg uncertainty principle

$$\Delta x\cdot \Delta p\ge \frac{\hslash }{2}$$

You cannot simultaneously precisely know position and momentum—perfect prediction impossible.

But determinists rebut: Uncertainty is just randomness, not freedom.

Chaos Amplification

Second limitation: Quantum chaos

Tiny quantum fluctuations, exponentially amplified through chaotic systems:

$$\Delta x(t)\sim \Delta x(0)\cdot e^{{\lambda }_{L}t}$$

Even if initial error extremely small, quickly becomes macroscopically unpredictable.

Brain is chaotic system—nonlinear dynamics of neural networks.

Butterfly effect: Quantum fluctuation of one neuron may change your decision.

But determinists rebut again: Chaos is also deterministic, just sensitive.

Self-Referential Undecidability

Third limitation (most profound): Undecidability of self-referential systems

If you try to predict your own decision:

• Your prediction affects your decision
• After decision changes, prediction fails
• Update prediction, decision changes again
• Infinite recursion

Corollary of Gödel’s incompleteness theorem: Self-referential systems cannot completely self-predict.

You cannot predict your entire future, even with perfect information—because prediction behavior itself changes future.

This isn’t ignorance, but logical necessity.

Act V: Levels of Freedom

Freedom is not binary (yes/no), but spectrum.

Level 0: No Freedom

• Stone: $I(M\to Y)=0$ (no action affects future)
• Pendulum: Completely deterministic, no choice space

Level 1: Reactivity

• Thermostat: $I>0$ but very small (on/off two states)
• Simple reflex: Stimulus → fixed reaction

Has causal influence, but choice space almost zero.

Level 2: Adaptability

• Bacterial chemotaxis: Adjust movement direction based on gradient
• ${\mathcal{E}}_k$ small but non-zero (multiple strategies)

Limited choice space, simple optimization.

Level 3: Learning

• Animal learning: Improve strategies through trial and error
• $I(M\to Y)$ increases (actions affect future environment)

Choice space expands with experience.

Level 4: Planning

• Tool use: Primates, birds, octopuses
• Predict future, plan multi-step actions
• ${\mathcal{E}}_k$ significant ($k$-step lookahead)

Long-term causal influence.

Level 5: Meta-Cognition

• Humans: Think about thinking
• Evaluate reasons for choices
• Change own decision rules

Highest level: Choose how to choose.

$$I(M_{\text{meta}}\to M_{\text{object}})>0$$

You not only choose actions, but choose decision rules—this is meta-freedom.

Act VI: Physical Basis of Compatibilism

Now we can understand why compatibilism makes sense.

Frankfurt Case

Philosopher Frankfurt proposed:

You want to do $A$, and you do $A$. But if you wanted to do $B$, a controller would force you to do $A$.

Do you have freedom?

• Incompatibilists say: No (no other possibility)
• Compatibilists say: Yes (you act according to your will)

GLS Answer

Key is distinguishing:

Actual freedom: Future you can actually change Counterfactual freedom: If you did something else, what would happen

Physics only cares about actual: $I(M\to Y)$

If your actual action $M$ affects actual future $Y$ ($I>0$), you have actual freedom.

As for “if you chose $B$”—this is counterfactual, not within scope of physical measurement.

Compatibilism is right: As long as your actions truly change future, you have freedom—even if there’s a controller in other possible worlds.

Act VII: Enhancing Your Freedom

If freedom is $I(M\to Y)$ and ${\mathcal{E}}_k$, how to enhance freedom?

Strategy 1: Expand Information Channels

$$I(M\to Y)\le \text{Channel Capacity}$$

Your freedom is limited by capacity of “action → observation” channel.

Enhancement methods:

• Learn skills: Increase available action space (learn driving, programming, instruments…)
• Expand perception: Increase observation dimensions (learn new languages, understand new fields…)

Skills = expanding causal influence.

Strategy 2: Extend Time Horizon

$${\mathcal{E}}_k=\max I(A^k;S_{k+1})$$

Larger $k$, farther lookahead, higher degree of freedom.

Enhancement methods:

• Planning ability: Consider long-term consequences (saving, learning, health investment…)
• Delayed gratification: Resist immediate temptations, pursue long-term goals

Patience = freedom across time.

Strategy 3: Reduce Constraints

Freedom is limited by:

$$I(M\to Y|\text{Constraints})$$

More constraints, less freedom.

Reduction methods:

• Economic freedom: Escape poverty (more choice space)
• Health freedom: Maintain physical health (action capability)
• Cognitive freedom: Overcome biases, fears (psychological constraints)

Constraints are cages of freedom.

Strategy 4: Increase Energy Supply

Remember energy constraint:

$$⟨W⟩+k_{B}T⟨I⟩\ge \Delta F$$

More energy = more information processing = more freedom.

Methods:

• Adequate sleep: Restore brain energy
• Balanced nutrition: Provide glucose
• Physical exercise: Improve metabolic efficiency

Fatigue reduces freedom—your decision quality declines.

Act VIII: Moral Responsibility

If freedom has degree, how to define moral responsibility?

Traditional Dilemma

• If humans have no freedom, how to hold responsible? (“He couldn’t help it”)
• If humans have absolute freedom, why are genes, environment so influential?

GLS Answer

Responsibility proportional to $I(M\to Y)$:

$$\text{Responsibility}\propto I(M\to \text{Outcome})$$

Your responsibility for outcome equals causal contribution of your actions to outcome.

This explains:

• Children have lighter responsibility: $I$ small (cognition immature, weak lookahead ability)
• Mental illness reduces responsibility: $I$ reduced by pathological constraints
• Coercion/threat reduces responsibility: External constraints compress choice space

Responsibility is continuous, proportional to causal influence—this matches legal practice (like “diminished responsibility”).

Meaning of Punishment

If freedom is limited, does punishment still have meaning?

GLS answer: Punishment is information feedback, enhancing future $I(M\to Y)$.

• Deterrence: Increase expected cost of crime, change decision function
• Rehabilitation: Provide new information, expand choice space
• Isolation: Temporarily reduce $I(M\to Y)$ (reduce social influence)

Punishment isn’t revenge, but system optimization—adjusting distribution of individual causal influence.

Act IX: Collective Freedom

Freedom is not only individual property, but also collective property.

Society’s Degree of Freedom

$$I_{\text{collective}}=I(M^{\text{society}}\to Y^{\text{society}})$$

Society’s freedom = influence of collective actions on collective future.

This depends on:

• Information flow: Freedom of speech, press freedom (expand diversity of $M$ and $Y$)
• Power dispersion: Democratic systems (increase number of effective actors)
• Education: Enhance individual ${\mathcal{E}}_k$ (collective wisdom)

Problem with authoritarian societies: Information concentrated in few, $I_{\text{collective}}$ very small—society loses adaptive capacity.

Technology and Freedom

How does technology affect freedom?

Double-edged sword:

Enhance freedom:

• Internet: Information acquisition cost ↓, $I(M\to Y)$↑
• Automation: Free time, choice space↑
• AI assistance: Cognitive ability↑, ${\mathcal{E}}_k$↑

Limit freedom:

• Algorithmic recommendations: Shrink information window, $I$↓
• Surveillance: Action space constrained
• Addictive design: Hijack decision systems

Net effect of technology depends on design—does it expand or compress human $I(M\to Y)$?

Act X: Ultimate Freedom

Finally, philosophical question:

Can You Choose Your Own Goals?

Traditional free will discussion: You can choose actions.

Deeper question: Can you choose desires themselves?

• You want sweets (genetically determined)
• You want recognition (evolutionarily shaped)
• You want to understand world (culturally instilled?)

If desires aren’t chosen by you, is your “freedom” real?

Meta-Freedom

GLS answer: You have limited meta-freedom—can partially change objective function.

$$I(M_{\text{meta}}\to \text{Goals})>0$$

Evidence:

• Meditation can change emotional responses
• Cognitive behavioral therapy can change beliefs
• Philosophical reflection can change values

You cannot completely rewrite brain (constrained by biology), but can adjust at margins—this is meta-freedom.

Highest Freedom: Accepting Constraints

Stoicism says: Freedom is not absence of constraints, but understanding constraints and choosing within them.

$$\text{True Freedom}=\underset{M\in \text{Possible}}{\max }I(M\to Y|\text{Understanding Constraints})$$

Freedom isn’t “do whatever you want,” but “maximize causal influence within possible range”.

You cannot fly (physical constraint), but can choose:

• Learn flying (pilot aircraft)
• Learn skydiving (brief flying sensation)
• Learn physics (understand why can’t fly)

Accept constraints, optimize within constraints—this is mature freedom.

Take-Home Thoughts

Right now, you’re reading this article.

You can choose:

• Continue reading (action $M_1$)
• Stop and think (action $M_2$)
• Close page (action $M_3$)

Is this choice real?

Physics says: Yes—as long as your choice can change your future $(I(M\to Y)>0)$.

If you continue reading, you’ll gain more information, change thinking patterns, affect future behavior. If you stop, you’ll maintain current cognitive state.

$$I(\text{finish article}\to \text{future you})\ne I(\text{don’t read}\to \text{future you})$$

This inequality is your freedom.

And this freedom has cost:

• Your brain is consuming glucose (energy cost)
• Your attention is focusing (opportunity cost)
• Your neural networks are adjusting (information processing cost)

You pay for freedom—every choice burns thermodynamic free energy.

But this also means: Your choices are real, have physical consequences, can change the world.

Freedom is not illusion—it’s physical reality with cost.

Next (Final): “The Geometric Map of Meaning: Information Geometry of Value, Purpose, and Existence”

We will see that ethical values, meaning of life, purpose of existence are neither emptiness nor subjective, but objective structures on information geometric manifolds.